Stabilizers of Classes of Representable Matroids

Let M be a class of matroids representable over a field F. A matroid N # M stabilizes M if, for any 3-connected matroid M # M, an F-representation of M is uniquely determined by a representation of any one of its N-minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N, or are obtained by a single-element extension followed by a single-element coextension. This result is used to prove that a 3-connected quaternary matroid with no U3, 6 -minor has at most (q&2)(q&3) inequivalent representations over the finite field GF(q). New proofs of theorems bounding the number of inequivalent representations of certain classes of matroids are given. The theorem on stabilizers is a consequence of results on 3-connected matroids. It is shown that if N is a 3-connected minor of the 3-connected matroid M, and |E(M)&E(N)| 3, then either there is a pair of elements x, y # E(M) such that the simplifications of M x, M y, and M x, y are all 3-connected with N-minors or the cosimplifications of M"x, M"y, and M"x, y are all 3-connected with N-minors, or it is possible to perform a 2&Y or Y&2 exchange to obtain a matroid with one of the above properties. 1999